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compute.py
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from sympyplus import *
# functions that give the individual terms
def termL1(grad1,grad2):
return innerp(grad1,grad2)
def termL2(grad1,grad2):
term = 0
for ii in range(5):
for jj in range(5):
term += (grad1.row(ii) * E[ii]).dot(E[jj] * grad2.row(jj).T)
return term
def termL3(grad1,grad2):
term = 0
for ii in range(5):
for jj in range(5):
term += (grad1.row(ii) * E[jj]).dot(E[ii] * grad2.row(jj).T)
return term
def term_twist_var(q,p):
""" Returns the variational derivative of the twist term """
Q = QTensor(q)
P = QTensor(p)
return 2*c.q0*mixedp(Q,P) + 2*c.q0*mixedp(P,Q)
# compute function, which puts the PDE system together
def compute():
from q3d.config import constants as c
from q3d.config import settings
# set up Qvector objects
nu = AbstractVector('nu')
q = QVector('q')
Dq = q.grad
Dqq = Param([Dq,q])
Q = q.tens
p = QVector('p')
Dp = p.grad
Dpp = Param([Dp,p])
P = p.tens
r = QVector('r')
Dr = r.grad
Drr = Param([Dr,r])
R = r.tens
qp = QVector('q_prev')
Dqp = qp.grad
QP = qp.tens
qpp = QVector('q_prev_prev')
Dqpp = qpp.grad
QPP = qpp.tens
qnp = QVector('q_newt_prev')
Dqnp = qnp.grad
Dqnpqnp = Param([Dqnp,qnp])
QNP = qnp.tens
f = QVector('f')
g = QVector('g')
Q0 = c.S0*(outerp(nu,nu) - (1.0/3.0)*eye(3))
Pi = eye(3) - outerp(nu,nu)
# Define 'energies' used to calculate the energy
energies = EnergyForm(Dqq,Dpp,Drr)
energies.add_domain(GeneralForm(c.L1/2*termL1(Dq,Dq)+c.L2/2*termL2(Dq,Dq)+c.L3/2*termL3(Dq,Dq),Dqq,name='Elastic Energy'))
if settings.pde.formulation == 'default':
energies.add_domain(GeneralForm(2*c.q0*c.L1*mixedp(Q,Q),Dqq,name='Twist Energy'))
elif settings.pde.formulation == 'lavrentovich':
energies.add_domain(GeneralForm(2*c.q0*c.L1*mixedp(Q,Q) + 2*c.q0**2*c.L1*innerp(Q,Q),Dqq,name='Twist Energy'))
energies.add_domain(GeneralForm((1/c.ep**2)*(1 - (c.A/2)*innerp(Q,Q) - (c.B/3)*trace(Q**3) + (c.C/4)*trace(Q**2)**2),Dqq,name='Bulk Energy'))
energies.add_domain(GeneralForm(-f.dot(q),Dqq,name='Domain Forcing Energy'))
energies.add_boundary(GeneralForm(c.W0/2*innerp(Q-Q0,Q-Q0),Dqq,name='Homeotropic Anchoring'))
energies.add_boundary(GeneralForm(c.W1/2*innerp(Q-Pi*Q*Pi+c.S0/3*outerp(nu,nu),Q-Pi*Q*Pi+c.S0/3*outerp(nu,nu))+c.W2/4*(innerp(Q,Q)-2*c.S0**2/3)**2,Dqq,name='Planar-degenerate Anchoring'))
energies.add_boundary(GeneralForm(-g.dot(q),Dqq,name='Boundary Forcing Energy'))
energies_minmom = energies.copy()
energies_minmom.add_domain(GeneralForm(1/(2*c.dt) * innerp(Q-QP,Q-QP),Dqq,name='Min Moments Energy'))
# Construct PDE
lhs_d = [ # lhs domain
GeneralForm(c.L1*termL1(Dq,Dp)+c.L2*termL2(Dq,Dp)+c.L3*termL3(Dq,Dp),Dqq,Dpp,name='a_E(Q,P)'),
GeneralForm(2*c.q0*c.L1*(mixedp(Q,P) + mixedp(P,Q)),Dqq,Dpp,name='a_T(Q,P)'),
GeneralForm((1/c.ep**2)*(-c.A*innerp(Q,P) - c.B*innerp(Q**2,P) + c.C*innerp(Q,Q)*innerp(Q,P)),Dqq,Dpp,name='Dψ(Q):P')
]
if settings.pde.formulation == 'lavrentovich':
del lhs_d[1]
lhs_d.append(GeneralForm(2*c.q0*c.L1*(mixedp(Q,P) + mixedp(P,Q)) + 4*c.q0**2*c.L1*innerp(Q,P),Dqq,Dpp,name='a_T(Q,P)'))
rhs_d = [ # rhs domain
GeneralForm(f.dot(p),Dpp,name='f(P)')
]
lhs_b = [ # lhs boundary
GeneralForm(c.W0*q.dot(p),Dqq,Dpp,name='W0(Q:P)'),
GeneralForm(c.W1*innerp(Q-Pi*Q*Pi,P),Dqq,Dpp,name='W1(Q-ΠQΠ):P'),
GeneralForm(c.W2*((innerp(Q,Q) - 2*c.S0**2/3)*innerp(Q,P)),Dqq,Dpp,name='(|Q|^2-2S0^2/3)(Q:P)')
]
rhs_b = [ # rhs boundary
GeneralForm(c.W0*innerp(Q0,P),Dpp,name='W0(Q0:P)'),
GeneralForm(c.W1*innerp(-c.S0/3*outerp(nu,nu),P),Dpp,name='(-S0/3(nu⊗nu):P)'),
GeneralForm(g.dot(p),Dpp,name='g(P)')
]
if settings.pde.grad_desc:
lhs_d.append(GeneralForm((1/c.dt)*(1/c.ep**2)*q.dot(p),Dqq,Dpp,name='(Ω)Q:P/dt'))
lhs_b.append(GeneralForm((1/c.dt)*(c.W2)*q.dot(p),Dqq,Dpp,name='(Γ)Q:P/dt'))
rhs_d.append(GeneralForm((1/c.dt)*(1/c.ep**2)*qp.dot(p),Dpp,name='(Ω)Q_p:P/dt'))
rhs_b.append(GeneralForm((1/c.dt)*(c.W2)*qp.dot(p),Dpp,name='(Γ)Q_p:P/dt'))
# assemble two PDEs
pde_d = PDE(lhs_d,rhs_d,Dqq,Dpp,over='domain')
pde_b = PDE(lhs_b,rhs_b,Dqq,Dpp,over='boundary')
# apply newton's method
pde_nm_d = pde_d.newtons_method(Dqnpqnp,Dpp,Dqq)
pde_nm_b = pde_b.newtons_method(Dqnpqnp,Dpp,Dqq)
# to make PDE system with positive definite lhs, remove non positive definite parts
pde_pd_d = pde_nm_d.copy()
pde_pd_d.rmv_lhs_form('∂(Dψ(Q):P)')
pde_pd_d.rmv_lhs_form('∂(a_T(Q,P))')
# Create relevant UFL strings
out = {
'pde_nm_d' : pde_nm_d.ufl,
'pde_nm_b' : pde_nm_b.ufl,
'energies' : energies.ufl_dict,
'energies_minmom' : energies_minmom.ufl_dict,
'pde' : [pde_d.ufl, pde_b.ufl],
'pde_nm' : [pde_nm_d.ufl, pde_nm_b.ufl],
'pde_pd' : [pde_pd_d.ufl, pde_nm_b.ufl],
}
return out
# END OF CODE